Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 16 / 2 / a / Solution

The convention intended by this question is existence of a dense forward orbit, also called point transitivity. On a nonempty compact metric space, this means that some $x\in X$ satisfiesEquivalently, that forward orbit visits every nonempty open set.

The other common definition of topological transitivity uses intersections of iterated open sets, as in part (b). The distinction matters on spaces with isolated points: parts (b) and (c) ask precisely how the two conventions are related. We use point transitivity for the dense-orbit convention throughout this solution. As usual for this definition, the state space is assumed nonempty; on an empty space the open-set condition is vacuous and there is no point with a forward orbit.